An experiment is a procedure that can be repeated and whose observed outputs are uncertain.

The observed outputs of experiments are outcomes.

The set of possible outcomes is the sample space Ω.\Omega.

A subset of outcomes A⊆ΩA\subseteq\Omega is called an event, and an event occurs if, when an experiment is performed, the outcome ω∈Ω\omega\in\Omega satisfies ω∈A.\omega\in A.

The null event ∅\varnothing and sample space Ω\Omega are both events. The null event will never occur and the sample space event will always occur.

The singleton subsets of Ω\Omega which are subsets that contain exactly one outcome from Ω\Omega are called the elementary events of Ω.\Omega.

Events AA and BB are disjoint events if and only if A∩B=∅.A \cap B = \varnothing. A collection of events E1,E2,…E_1, E_2, \ldots is pairwise disjoint if Ei∩Ej=∅  ∀ i≠j.E_i \cap E_j = \varnothing \,\, \forall \, i \neq j.

The set of all subsets of Ω\Omega is the power set P(Ω).\mathcal{P}(\Omega). All events are subsets of Ω\Omega however not all subsets of Ω\Omega (elements of P(Ω)\mathcal{P}(\Omega)) are necessarily events. Hence, the set of all possible events F\mathcal{F} is a subset of P(Ω),\mathcal{P}(\Omega), F⊆P(Ω).\mathcal{F}\subseteq\mathcal{P}(\Omega).

If the set of events F\mathcal{F} has the following properties:

  • if events A,B∈FA,B\in\mathcal{F} then A∪B∈FA\cup B\in\mathcal{F}
  • if event A∈FA\in\mathcal{F} then A‾∈F\overline{A}\in\mathcal{F}
  • the null event is in F,\mathcal{F}, ∅∈F\varnothing\in\mathcal{F}

where A‾:=Ω∖A\overline{A} := \Omega \setminus A is the complement of A,A, then F\mathcal{F} is an algebra of sets. An algebra is closed under finite unions and finite intersections.

An algebra F\mathcal{F} is a σ\sigma-algebra if it satisfies the following properties:

  • if events E1,E2,…∈FE_1,E_2,\ldots\in\mathcal{F} then
⋃i=1∞Ei∈F\bigcup_{i=1}^\infty E_i\in\mathcal{F}
  • if event A∈FA\in\mathcal{F} then A‾∈F\overline{A}\in\mathcal{F}
  • the null event is in F,\mathcal{F}, ∅∈F.\varnothing\in\mathcal{F}.

The elements in a σ\sigma-algebra are measurable subsets of the set it is defined on. An ordered pair containing a set and a σ\sigma-algebra of its measurable subsets is called a measurable space.

Any experiment has an associated measurable space (Ω,F)\left(\Omega,\mathcal{F}\right) where Ω\Omega is the set of all possible outcomes and F\mathcal{F} is a σ\sigma-algebra of subsets of Ω\Omega which contains all possible events.